Under consideration for publication in Math. Proc. Camb. Phil. Soc.

1

The fine Tate-Shafarevich group
By Christian WuthrichSéction de mathématiques, CSAG
École Polytechnique Fédérale
1015 Lausanne
Switzerland(Received)

Abstract
Within the Tate-Shafarevich group of an elliptic curve E defined over a number field
K, there is a canonical subgroup defined by imposing stronger conditions at the places
above a given prime p. This group appears naturally in the Iwasawa theory for E. We
propose a study of what one can say about the relation to the full Tate-Shafarevichgroup. Some numerical examples are included, as well as a few conjectures.

1. Introduction
Let E be an elliptic curve defined over a number field K. We fix a prime number
p. Everything will depend on p, but we will omit if from our notations. For an abeliangroup A, the p-primary part of A will be denoted by A(p). The p-primary part of theTate-Shafarevich group X(E/K)(p) is defined to be the kernel of the localisation map
0 X(E/K)(p) H 1 (K, E)(p) ⊕υ H 1 (Kυ , E)(p).

We will define in section 2 a certain canonical subgroup of X(E/K)(p) by imposing
stronger conditions at the places above p. It will be called the fine Tate-Shafarevichgroup, denoted by another Russian letter (E/K). It is actually defined as the union of asequence of subgroups k (E/K) in X(E/K)[pk ] and it represents the ”Tate-Shafarevich
part“ of the fine Selmer group R(E/K) as defined in [CS05]. See section 2 for the details.
The motivation for studying this group comes from the Iwasawa theory of E. Itsbehaviour is closely linked to Euler systems for the Tate-module Tp E = lim E[pk ], like←−the one by Kato [Kat04]. Suppose E is defined over Q. If the L-series of E does notvanish at 1, then Kato’s Euler system c is such that cQ ∈ H 1 (Q, Tp E) is not torsion.
By theorem 2.2.3 in [Rub00], the fine Selmer group R(E/Q), which is then equal to(E/Q), is finite. Moreover the divisibility of cQ by p may be used to give an upperbound on the size of (E/Q).
For more details on the Iwasawa theory of the fine Selmer group, we refer the reader
to [Wut05], [Wut04] and [PR95].
This article focuses on the analysis of the fine Tate-Shafarevich group over the number
field K. The first part concerns its relation to the full Tate-Shafarevich group. For K = Q,
we prove in Theorem 3·4 that (E/Q) is non-trivial exactly if X(E/Q) is non-trivial,
and that they are equal if the rank of the curve is positive (see Theorem 3·5). NumericalChristian Wuthrich

2

examples are given in section 4 for curves of rank 0 and p = 2. We find three examples ofcurves E defined over Q with X(E/Q)(2) = Z/2Z ⊕ Z/2Z which show that all possible orders(namely 1,2 or 4) can occur for 1 (E/Q). The fine Tate-Shafarevich groups (E/Q)
have order 2, for the first two curves, and 4 for the last example.
We present a version of the Cassels-Tate pairing due to Flach [Fla90] for our subgroup (E/K) in section 6. Then the fine version of the Mordell-Weil group E(K),also obtained by imposing conditions at the places above p, is considered. We end thearticle with some conjectures on the fine Tate-Shafarevich group (E/K). In contrast
to the situation over Q, it is not true for larger fields that the fine subgroup of the TateShafarevich group is often relatively big. In fact, for Zp -extensions it is conjectured thatthe growth of the fine Tate-Shafarevich group is significantly smaller.
Acknowledgements. It is a pleasure to thank John Coates, Mike Shuter, Paola Argentin
and Sylvia Guibert for their help and comments.
2. Definitions
For any k ≥ 1, we have a short exact sequence coming from the pk -descent E(K)/pk E(K)0κ

 S k (E/K)

 X(E/K)[pk ]

0

(2·1)

where S k (E/K) is the Selmer group defined as the kernel of the map
M
H 1 (K, E[pk ])
H 1 (Kυ , E)[pk ]υwith the sum running over all places υ in K. This Selmer group contains a subgroup
Rk (E/K), which we will call the fine Selmer group, obtained by imposing stronger conditions at the places above p, i.e. the sequenceM Rk (E/K)
 S k (E/K)

0H 1 (Kυ , E[pk ])υ|pkis exact. The subgroup R (E/K) is sometimes called the “strict” or “restricted” Selmer
group, but in order to avoid possible confusions with different definitions, we prefer theterminology in [CS05]. Similarly we can define a fine subgroup of the Mordell-Weil group.
Namely the following kernel
M
E(Kυ )/pk E(Kυ ),(2·2)
 M k (E/K)
 E(K)/pk E(K)

0
υ|p

k

which can also be written as the intersection of R (E/K) with the image of the Kummer
map κ in (2·1) inside S k (E/K).
Now, we can proceed to define the group k (E/K) as the quotient of Rk (E/K) by
M k (E/K). We call it the fine Tate-Shafarevich group. So we have an exact sequence0

 M k (E/K)κ

 Rk (E/K)

 k (E/K)

0

(2·3)

similar to the Kummer sequence (2·1). In order to prove that it is really a subgroup of
the Tate-Shafarevich group, we consider the diagram0 E(K)/pk E(K)

 S k (E/K)

 X(E/K)[pk ]

0

0

g
 ⊕E(Kυ )/pk E(Kυ )g1
 ⊕H (Kυ , E[pk ])0g ⊕H 1 (Kυ , E)[pk ]0The fine Tate-Shafarevich group3(the ⊕ stands for the sum over all places υ above p). We can extract an exact sequence0

 M k (E/K)

 Rk (E/K)

 X(E/K)[pk ]

δk

 Ck

(2·4)

where C k is the cokernel of the left vertical map in the above diagram. As a consequence,
one can view k (E/K) as a subgroup of X(E/K)[pk ] with quotient inside C k .
LimitsAs usual, it is interesting to consider the limits as k tends to infinity. We write S(E/K)
for the inductive limit lim S k (E/K) obtained from the inclusion E[pk ] ⊂  E[pk+1 ]. We−→
 E[pk ]denote the projective limit lim S k (E/K) induced from the map [p] : E[pk+1 ]←−kkby S(E/K). The same limits can be used for the subgroups R (E/K) and M (E/K).
S(E/K) = lim S k (E/K)−→
R(E/K) = lim Rk (E/K)−→
M(E/K) = lim M k (E/K)−→

S(E/K) = lim S k (E/K)←−
R(E/K) = lim Rk (E/K)←−
M(E/K) = lim M k (E/K)←−

For the fine Tate-Shafarevich group, we will denote the limit lim k (E/K) simply by−→(E/K). It is a subgroup of the p-primary part X(E/K)(p) of the Tate-Shafarevichgroup. We have two exact sequences0 M(E/K) R(E/K) (E/K)00 M(E/K) R(E/K) lim k (E/K)←−

0since all terms in (2·3) are finite. Both lim k (E/K) and Tp (E/K) = lim [pk ] are←−←−subgroups of Tp X(E/K), where, for an abelian group A, we write Tp A for the projective
limit lim A[pk ]. Of course, it is widely believed that these three groups are trivial. One←−
can show that
Lemma 2·1. The group lim k (E/K) has finite index in Tp (E/K).
←−
A proof can be found in [Wut04, Lemma II.1].
Lemma 2·2. The maps
1 (E/K) 2 (E/K)

 ...

are injective.
Proof. This is obvious from the following diagram. Here C k is as in (2·4).
0 k (E/K)δk X(E/K)[pk ]

 Ck

∩

0

g
k+1
  (E/K)g
 X(E/K)[pk+1 ]

(2·5)
δ

k+1

g
 C k+1

where the map on the right hand side is induced by the maps[p] : E(Kυ )/pk E(Kυ ) E(Kυ )/pk+1 E(Kυ ).

The following lemma tells us in some cases when the above sequence stabilises.

Christian Wuthrich

4Lemma 2·3. Let E be an elliptic curve over Q of rank 0 with E(Qp )[p] = 0. If pk kills
all elements in X(E/Q)(p), thenk (E/Q) = (E/Q).
Proof. Under the hypothesis on k, the map in the middle of (2·5) becomes an isomorphism. Because E(Q)/pk E(Q) = 0, the group C k is equal to E(Qp )/pk E(Qp ). Hence the C k+1 is injective if E(Qp )[p] = 0.
map induced by [p] on C k
Even with weaker assumptions, it is usually easy to verify for which k the sequence ofk (E/K) stabilises.
RemarkInstead of imposing stronger conditions only at places above p, we could more generally
consider a finite set Σ of places in K, containing the places above p, and then demandthat elements in S k (E/K) have trivial localisation in the group H 1 (Kυ , E[pk ]) for allplaces in Σ. We would obtain even smaller subgroups RΣk (E/K) and kΣ (E/K). Note thatRΣ (E/K) = lim RΣk (E/K) does not differ from R(E/K) because the groups E(Kυ )⊗Qp/Zp−→
are trivial for all places υ - p.
3. Comparison
Lemma 3·1. Let Kυ be a finite extension of Qp of degree nυ . Then
# E(Kυ )/pk E(Kυ ) = pnυ ·k · # E(Kυ )[pk ] .
Proof. Note that, if A is a subgroup of an abelian group B of finite index, then#B/pk B
#A/pk A=,k
#A[p ]#B[pk ]
provided all terms are well-defined. Denote by E ◦ (Kυ ) the Kυ -rational points on the
b the formal group associated toconnected component of the Néron-model of E and by E◦b a ), for any power ofE (see [Sil96]). Since E(Kυ ) has finite index in E (Kυ ) and in E(mυthe maximal ideal mυ in the ring of integers in Kυ , we obtain thatb υ )/pk E(mb υ)#E(Kυ )/pk E(Kυ )#E ◦ (Kυ )/pk E(Kυ )#E(m==b υ )[pk ]#E(Kυ )[pk ]#E ◦ (Kυ )[pk ]#E(m=b a )/pk E(m
b a)
#E(m
υ
υ
b a )[pk ]#E(mυb a ) to ma
Now, if a is large enough, the formal logarithm gives an isomorphism from E(m
υ
υ
a
k
a
kb
a
knυ
b
b
and we have E(mυ )[p ] = 0 and #E(mυ )/p E(mυ ) = p .
Proposition 3·2. The index of k (E/K) inside X(E/K)[pk ] is bounded byhiY
X(E/K)[pk ] : k (E/K) ≤ p[K:Q]·k ·#E(Kυ )[pk ].
υ|pProof. We saw that the quotient of the two groups in question is contained in thecokernel C k of the map from E(K)/pk E(K) to ⊕E(Kυ )/pk E(Kυ ). The previous lemma
proves that the target of this map has size bounded by the right hand side of the formulain the proposition.

The fine Tate-Shafarevich group

5Corollary 3·3. If E is defined over Q and E(Qp )[p] = 0, thenhiX(E/Q)[p] : 1 (E/Q) ≤ p.
Theorem 3·4. Let E be an elliptic curve defined over Q. Suppose X(E/Q)(p) is
finite, then (E/Q) is non-trivial if and only if X(E/Q)(p) is non-trivial.
Proof. The injection of (E/Q) into X(E/Q)(p) has cokernel in C = lim C k which−→
is a quotient of E(Qp ) ⊗ Qp/Zp . But this last group is isomorphic to Qp/Zp . If X(E/Q)(p)is finite and non-trivial, then it is of the form A ⊕ A for some abelian p-group A by the
Cassels-Tate pairing. So X(E/Q)(p) can not be embedded into Qp/Zp .
Theorem 3·5. Let E be an elliptic curve defined over Q of positive rank, then we have(E/Q) = X(E/Q)(p).
Proof. The group C in the proof of the previous theorem is the cokernel of the localisab 2 Zp )⊗ Qp/Zp ,tion map E(Q)⊗ Qp/Zp
 E(Qp )⊗ Qp/Zp . The target of this map equals E(pwhich is isomorphic to Qp/Zp via the p-adic elliptic logarithm map. If the rank of E is positive, the localisation map is non-trivial: a sufficiently large multiple of any point of infiniteb 2 Zp ) and has non-zero logarithm. Since E(Qp ) ⊗ Qp/Zp
order in E(Q) will belong to E(pis divisible the map has to be surjective and so C = 0.
If the rank is zero, (E/Q) may be strictly smaller than X(E/Q)(p) as we will seein the numerical examples in the next section.
We see here that the fine Tate-Shafarevich group for an elliptic curve over Q tends tobe rather large in comparison with the whole Tate-Shafarevich group. This is likely to be
specific to Q as we will announce a conjecture for certain larger fields later in section 8.
4. Numerical examples
In this section we give four numerical examples of fine Tate-Shafarevich groups for
p = 2. The four curves E are all defined over Q. They all have rank 0, as can be verified
using Kolyvagin’s result or a 3-descent; in fact for each curve we have E(Q) = Z/2Z ⊕ Z/2Z .
Then we perform a complete 2-descent on E as explained in [Sil96, Proposition X.1.4].
The complete 2-descent proves for the first three examples that X(E/Q)[2] = Z/2Z ⊕Z/2Z . Hence the fine Tate-Shafarevich group 1 (E/Q) can have 1, 2 or 4 elements. The
examples are chosen as to show that all three cases can occur. Furthermore we can provethat the 2-primary part X(E/Q)(2) has order 4 using a 4-descent or the Cassels-Tatepairing implemented in magma. So the fine Tate-Shafarevich group (E/Q) can have
either 2 or 4 elements by Corollary 3·3.
The details of the computations are only included for the first example.
An example with a trivial 1 (E/Q)
Let E/Q be the elliptic curve O3 of conductor 930 in Cremona’s table [Cre97]E:

y 2 + xy = x3 − 190 220 x − 10 0270 200

The Mordell-Weil group E(Q) is generated by two points T1 = (−80, 40) and T2 =321(− 321
4 , 8 ) of order 2. The analytic order of X(E/Q) is 4.
Let us change the equation to the non-minimal equationy 2 = (x + 320) · (x + 321) · (x − 640)

Christian Wuthrich

6

in which T1 has coordinates (−320, 0) and T2 is equal to (−321, 0). Let Σ be the set{2, 3, 5, 31} which includes all primes of bad reduction for E. The group Q(Σ, 2) is defined
to be the group of non-zero integers only divisible by elements in Σ modulo squares; whichin our case is isomorphic to (Z/2Z )5 . The above choice of a basis of E(Q)[2] allows us to
identify the Selmer group S 1 (E/Q) with a subgroup of the square of Q(Σ, 2). In fact, theimage of T1 is represented by (−15, 1) while T2 is sent to (−1, 1) in Q(Σ, 2)2 . One sees
immediately that T1 belongs to the fine Mordell-Weil group M 1 (E/Q) since both −15
and 1 are squares in Q×
2 , in other words there is a 4-torsion point
S1 = (23 + 27 + 28 + O(210 ), 23 + 28 + 29 + O(210 ))defined over Q2 such that 2 · S1 = T1 .
Now the first Selmer group S 1 (E/Q) is generated by the image of T1 and T2 andthe elements (5, 1) and (2, 1) in Q(Σ, 2)2 . We can conclude that R1 (E/Q) is equal to(−15, 1) · Z/2Z = M 1 (E/Q), and hence that 1 (E/Q) is trivial.
We now use the diagram (2·5) to compute the whole of (E/Q). We need to compute
C k first. Denote by Φ(E)(Q2 ) the group of connected components of the Néron-modelover Q2 . For our example Φ(E)(Q2 ) is a cyclic group of order 4; the reduction typeis I4 . The torsion group of E(Q2 ) is Z/4Z S1 ⊕ Z/2Z T2 . Since S1 maps to the generatorof Φ(E)(Q2 ), we know that E(Q2 ) = E ◦ (Q2 ) ⊕ Z/4Z S1 with E ◦ being the connectedcomponent of the identity of the Néron model. The torsion point T2 lies in the first layerof the formal group in the minimal model, and so we have E(Q2 ) = Z2 ⊕ Z/4Z S1 ⊕ Z/2Z T2 .
We conclude that C k is isomorphic to Z/2k Z ⊕ Z/2Z S1 . The map induced by [2] is theinjection Z/2k Z ⊂  Z/2k+1 Z on the first factor and the trivial map on the second:Ck/2k Z ⊕ Z/2Z S1

 C k+1Z(1,0)

 Z/2k+1 Z ⊕ Z/2Z S1The element (5, 1) in X(E/Q)[2] is mapped to the element δ 1 (5, 1) in C 1 representedbyQ1 = (2−2 + 1 + 26 + 27 + 29 + O(210 ), 2−3 + 2 + 22 + 23 + 26 + 27 + 29 + O(210 ))which belongs to the first factor of C 1 = Z/2Z ⊕ Z/2Z S1 . The image of the other generator(2, 1) can be represented by S1 . Hence the first element of X(E/Q)[2] always maps toa non-trivial element in C k and does therefore not belong to k (E/Q) for any k, while
the latter element has trivial image in C 2 and thus belongs to 2 (E/Q).
We can conclude the following:1 (E/Q) = 0 and 2 (E/Q) = (E/Q) = Z/2Z .
A 1 (E/Q) with 2 elements
Let E be the curve E5 of conductor 210
E:

y 2 + xy = x3 − 1200 050 x − 160 0200 000.

The Mordell-Weil group E(Q) is generated by two points T1 = (400, −200) and T2 =801
1(− 801
4 , 8 ) of order 2. A computation like for the first example shows that M (E/Q) = 0
1
1
Z
and that R (E/Q) =  (E/Q) = /2Z . Using that X(E/Q)(2) has 4 elements one showsthat k (E/Q) = (E/Q) = Z/2Z for all k ≥ 1.

The fine Tate-Shafarevich group

71

An example where  (E/Q) has 4 elements
Let E be the elliptic curveE:y 2 + xy = x3 − x2 − 30 861 x − 910 368.

It is labelled 1287E2 in the tables of Cremona. For this curve we have that M 1 (E/Q) = 0
and R1 (E/Q) = 1 (E/Q) = Z/2Z ⊕ Z/2Z = X(E/Q)[2]. Here we get (E/Q) = Z/2Z ⊕Z/2Z .
A bigger example
Finally, we include an example1of a larger Tate-Shafarevich group. The curve EE:

y 2 + x y = x3 − x2 − 2130 2300 796 x + 10 1830 71206880 107

of conductor 32 · 132 · 17 · 1572 is a twist of the curve 17A2. The Mordell-Weil group has
rank 0 and it has 4 rational 2-torsion points. The Birch and Swinnerton-Dyer conjecture
predicts the order of X(E/Q) to be 16.
This time, we find that S 1 (E/Q) = (Z/2Z )6 and so X(E/Q)[2] = (Z/2Z )4 . It can beverified as before that the 2-primary part of X(E/Q) consists of these 16 elements. The
computation of the fine Selmer group shows that R1 (E/Q) = 1 (E/Q) = (Z/2Z )3 . The
fine Tate-Shafarevich is equal to (E/Q) = (Z/2Z )3 .
5. Geometric interpretation
As usual, an element in the Selmer group and in the Tate-Shafarevich group may beinterpreted geometrically in terms of torsors of E. Let F be an extension of K. An p k  E defined over F such that there iscovering over F is a morphism of curves α : C
an isomorphism φ over F̄ having the property that the following diagram commutes.

α


≺

φ
E
C ...............................
[pk ]

E

Note that C is a torsor of E. Two p -coverings α and α0 are isomorphic if there is a φdefined over F such that
k



α

≺

φ

C

 C0
α0

E

k

commutes. In particular, a p -covering α is isomorphic to the trivial pk -covering givenby [pk ] : E
 E if and only if there is a point defined over F in the fibre of α above
O ∈ E(F ). It is well known that the elements of H 1 (F, E[pk ]) can be represented in aunique way as isomorphism classes of pk -coverings.
Proposition 5·1. The fine Selmer group Rk (E/K) is in bijection with the pointed
set of isomorphism classes of pk -covering α : C
 E over K such that(i) C(Kυ ) is non empty for all places υ in K and,1

I am grateful to Christophe Delaunay for helping me to find this example

Christian Wuthrich8(ii) for all places υ above p, there is a Kυ -rational point Pυ in the fibre α−1 (O) above
O ∈ E(K).
The point Pυ is only defined up to addition by a Kυ -rational pk -torsion point. For thefine Tate-Shafarevich group, we have the following description
Proposition 5·2. The fine Tate-Shafarevich group k (E/K) can be viewed as thepointed set of isomorphism classes of principal homogeneous spaces C under E definedover K such that(i) C(Kυ ) is non empty for all places,(ii) there is a K-rational effective divisor D of degree pk on C and,(iii) for all places υ above p, the divisor D contains a Kυ -rational point Pυ in its
support.
6. Duality
Cassels constructed a pairing on the Tate-Shafarevich group, whose kernel consists of
the infinitely divisible elements. It follows that a finite Tate-Shafarevich group has squareorder. The construction of Cassels was extended by Flach in [Fla90]. In order to state
the duality for the fine Selmer group, we need to define another generalised version of
the Selmer group, namely the group Q(E/K) defined by the exact sequenceYY
H 1 (Kυ , E(p))/ div .
H 1 (Kυ , E)(p) ×
 Q(E/K)
 H 1 (K, E(p))

0
υ|p

υ-p

Here we denote by A/ div the quotient of A by its maximal divisible subgroup wheneverA is a discrete abelian group. The enlarged Selmer group Q(E/K) contains the usual
Selmer group S(E/K). Let Q(E/K)(p) be the quotient of Q(E/K) by the image ofE(K) ⊗ Qp/Zp .
Theorem 6·1. There is a alternating pairing on R(E/K) × Q(E/K) with values in/Zp whose kernel on each side consists of the divisible elements. In particular this identifies R(E/K)/ div with the dual of Q(E/K)/ div.

Qp

This is just a reformulation of the main theoren in [Fla90] when using the local conditions
Wυ = 0 for all places υ.
From the Poitou-Tate sequence [NSW00, Theorem 8.6.13] we may deduce two otherfacts. Let Σ be any finite set of places containing the ones above p, and those where E
has bad reduction. The group R(E/K) is dual to the kernel ofM
H 2 (GΣ (K), Tp E)
H 2 (Kυ , Tp E)υ∈Σ

where GΣ (K) stands for the Galois group of the maximal extension of K which is unramified outside Σ. The target of the map is the dual of the finite group ⊕υ∈Σ E(Kυ )(p).
Furthermore there is an exact sequence
M0 R(E/K)
 H 1 (GΣ (K), E(p))

H 1 (Kυ , E(p))υ∈Σ

 H 1 (GΣ (K), Tp E)∧
Here M∧ H 2 (GΣ (K), E(p))

denotes the Pontryagin dual of M .

 0.

The fine Tate-Shafarevich group

9

7. The fine Mordell-Weil group
We now turn our interest to the other side of the fine Selmer group, namely the fineMordell-Weil group. In the limit, we have a compact version M(E/K) and a discrete
version M(E/K). Note that, even though the group E(K) ⊗ Qp/Zp is divisible, the group
M(E/K) is not necessarily divisible as we see in the following example. Let E be theelliptic curve 37A1 given by the equationE:y 2 + y = x3 − x(7·1)

defined over Q, whose Mordell-Weil group E(Q) is generated by the point of infinite orderP = (0, 0). We shall see that the fine Mordell-Weil group M(E/Q) for the prime p = 179
is a finite non-trivial group. The curve has good, ordinary, non-anomalous reduction atp. The point 81 · P is the first multiple of P that lies in the kernel of reduction modulo
p, but it even lies in the second layer of the formal group, i.e. the denominator of thex-coordinate is divisible by p4 . In other words, the point P is divisible by p in E(Qp ).
Hence M(E/Q) = Z/pZ .
More generally, the size of the group M(E/K)/ div measures the divisibility of thegenerators in E(Qp ). For curves of rank 1 over Q it can be expressed in terms of thep-adic logarithm of the generator. See lemma 9 in [CM94]; the first table in this article
can be used to find further examples like the one above.
If A is an abelian group, we denote by A? the p-adic completion lim A/pk A of A.
←−
Theorem 7·1. We have the following exact sequence M(E/K)0 Tp M(E/K)

T TorsZp (D) M(E/K)/ div

0

where D is the cokernel of the map E(K)?
 ⊕υ|p E(Kυ )? and T is the cokernel ofthe corresponding localisation map on the p-primary part, namely E(K)(p)   ⊕ υ|pE(Kυ )(p).
Proof. We split the exact sequence0 M(E/K) E(K) ⊗ Qp/Zp

 ⊕ E(Kυ ) ⊗ Qp/ZpC

0

by defining B to be the image of the middle map. Now, the middle terms are divisible
and so is B. This provides us with two exact sequences0 Tp M(E/K)
 Tp E(K) ⊗ Qp/Zp
 Tp B M(E/K)/ div 0 (7·2)0 Tp B ⊕ Tp (E(Kυ ) ⊗ Qp/Zp ) Tp C0(7·3)

We apply the snake lemma to the following diagram00 E(K)(p)g

 E(K)?

 Tp (E(K) ⊗ Qp/Zp )0g
 ⊕E(Kυ )(p)g
 ⊕E(Kυ )?

g ⊕Tp (E(Kυ ) ⊗ Qp/Zp )0Defining A to be the cokernel of the vertical map on the right hand side, we find thesequence0 M(E/K) Tp M(E/K)

T

D

A

0

containing the first four terms of the sequence in the theorem. Furthermore, by the twoChristian Wuthrich

10sequences involving (7·2) and (7·3), we see that the cokernel A is in the following sequence0 M(E/K)/ div

A Tp C 0.

The fact that Tp C is Zp -free proves that TorsZp (A) = M(E/K)/ div.
Note that if E is defined over Q and the rank is positive then D is torsion.
Corollary 7·2. If the curve E is defined over Q and E(Qp )[p] is trivial, then the
canonical map from M(E/Q) to Tp M(E/K) is an isomorphism.
Note that this does not hold in general, as we can see in the following example. Let E
be the curveE:y 2 + x y = x3 + 1(7·4)

of conductor 433 and let p = 3. The curve E has no non-trivial torsion point definedover Q, but two linearly independent points of infinite order, namely P1 = (−1, 0) andP2 = (0, 1). Over Q3 there is a 3-torsion point on E, namelyS = 2 + 2 · 33 + 36 + 2 · 37 + 39 + O(310 ), 2 · 32 + 33 + 35 + 2 · 38 + O(310 ) .
The reduction of E at p = 3 is good anomalous with 6 points on the reduced curvee 3 ). The reduction of the point P2 generates E(Fe 3 ). On the other hand the point
E(F
13 38
b
P1 + 2 · P2 = ( , ) is in the first layer of the formal group E(3Z
3 ). It follows that
927

 E(Q3 )? is surjective. On the other hand the group T is not trivial sinceE(Q)?
there is this point S of order 3 that it not defined over Q. From the sequence in theabove theorem, we conclude that the index of M(E/Q) in Tp M(E/Q) is equal to p.
As a consequence of the previous theorem, we find some more information on the fineMordell-Weil group

Corollary 7·3. The compact fine Mordell-Weil group M(E/K) is a free Z p -moduleof finite rank. If the rank r of the curve is zero then M(E/K) = M(E/K) = 0, otherwise,
if r is positive, thenr − [K : Q] ≤ corankZp M(E/K) = rankZp M(E/K) ≤ r − 1
If K = Q, we have equality.
Proof. The only thing that is left to prove is the left hand inequality. Let P be ab υ)
point of infinite order on E(K). A certain multiple of P will lie in all the kernels E(mb υ ) intoof reduction modulo all places above p. Since there is an injection from E(mE(Kυ )? for all υ | p, this multiple will not lie in M(E/K). (Compare to the proof oftheorem 3·5).
In particular, for curves of rank 0, the sequence in Theorem 7·1 becomes simply an
isomorphism between T and TorsZp (D). If E is defined over Q and has rank 1 then we
get a sequence
0

T

D

 M(E/Q)0of finite groups.
8. Conjectures
The purpose of this last section is to announce some conjectures on the finiteness ofthe fine Tate-Shafarevich group. Of course, there is a first basic

The fine Tate-Shafarevich group11

Conjecture 8·1. The fine Tate-Shafarevich group (E/K) of an elliptic curve overa number field K is finite. For a given E/K it is trivial for all but a finite number ofprimes p.
It is a weaker form of what should be believed by any gentleman namely that X(E/K)
is already finite. Note that over Q for a curve of positive rank the two statements areequivalent by theorem 3·5.
The original interest in this group comes from Iwasawa theory for elliptic curves. Let
E be an elliptic curve, for simplicity, defined over Q. Using the methods of PerrinRiou [PR95] one is able to compute the growth of the size of the fine Tate-Shafarevichgroup in certain towers of number fields. See [Wut05] and [Wut04] for more details. LetQ(µ[pn+1 ]) be the cyclotomic field of pn+1 -th roots of unity and let n Q be the uniquesubfield of Q(µ[pn+1 ]) with Galois group Gal(n Q : Q) isomorphic to Z/pn Z . The cyclotomic
Zp -extension ∞ Q of Q is defined to be the union of all n Q as n varies over the positive
integers.
Denote by en the integer such that #(E/n Q) = pen . By a famous theorem of
Kato [Kat04], we know that the integer en grows like µ · pn + λ · n + O(1) for some
integers µ > 0 and λ > 0. There is a first conjecture, a special case of conjecture A
in [CS05], due to Coates and Sujatha.
Conjecture 8·2. The growth of the order of (E/n Q) is of the form en = λ · n +O(1), i.e. µ = 0.
The striking thing about this conjecture is that a similar statement for X(E/n Q)
is known to be false ever since the beginnings of Iwasawa theory for elliptic curves,see [Maz72]. For our case of a curve over Q, the conjecture is verified if E admits an
isogeny of degree p defined over Q. This follows from corollary 3.5 in [CS05].
As an example, we might consider the curve of conductor 11 given byE:

y 2 + y = x3 − x2 − 70 820 x − 2630 580
n

for which it is known that the order of X(E/n Q)(5) for p = 5 grows like 52·5 , i.e. µ = 2and λ = 0. Nevertheless, the order of (E/n Q) stays bounded, see [Wut05].
The numerical evidence computed in [Wut04] suggest that it might be possible thatthe following question has a positive answer.
Question 8·3. Is the order of (E/n Q) bounded independently of n, i.e. do we havethat µ = λ = 0 ?
In other words, we could ask if (E/∞ Q) is finite. For the supersingular case the eventuality that the answer might be ”yes“ was mentioned to me by Kurihara and Pollack.
Nevertheless, there is one hint that it could be wrong in the extension to the situationof ordinary reduction. In the case of a global field of positive characteristic different fromp, say K is a function field of a curve over a finite field, the fine Tate-Shafarevich groupof an elliptic curve E over K naturally coincides with the full Tate-Shafarevich group,
since there are no places above p. Moreover it is equal to the Brauer-Grothendieck group
Br(E) of the elliptic surface E whose generic fibre is E, see [Tat95]. The analogue ofX(E/∞ Q) is the the Brauer group Br(Ē) of the surface E but over the algebraic closureof the finite base field. There are certainly many examples of infinite Br(Ē).

12

Christian Wuthrich

On the other hand, we know that Br(Ē) is co-finitely generated over Zp and so theconjecture 8·2 is known for function fields.
But at least, we can be sure that a weaker form of the question should have a positive
answer, namely
Conjecture 8·4. For all but a finite number of primes p, the order of (E/n Q) isbounded independently of n.
Compare with the conjectures in [Wut05] on the density of such primes.
REFERENCES[CM94] John Coates and Gary McConnell, Iwasawa theory of modular elliptic curves of analytic rank at most 1, J. London Math. Soc. (2) 50 (1994), no. 2, 243–264.
[Cre97] John E. Cremona, Algorithms for modular elliptic curves, second ed., Cambridge
University Press, 1997.
[CS05] John Coates and Ramdorai Sujatha, Fine Selmer groups of elliptic curves over p-adic
Lie extensions, Math. Ann. 331 (2005), no. 4, 809–839.
[Fla90] Matthias Flach, A generalisation of the Cassels-Tate pairing, J. Reine Angew. Math.
412 (1990), 113–127.
[Kat04] Kazuya Kato, p-adic Hodge theory and values of zeta functions of modular forms,
Cohomologies p-adiques et application arithmétiques. III, Astérisque, vol. 295,
Société Mathématique de France, Paris, 2004.
[Maz72] Barry Mazur, Rational points of abelian varieties with values in towers of number
fields, Invent. Math. 18 (1972), 183–266.
[NSW00] Jürgen Neukirch, Alexander Schmidt, and Kay Wingberg, Cohomology of number
fields, Grundlehren der Mathematischen Wissenschaften, vol. 323, Springer, 2000.
[PR95] Bernadette Perrin-Riou, Fonctions L p-adiques des représentations p-adiques, Astérisque (1995), no. 229, 198.
[Rub00] Karl Rubin, Euler systems, Annals of Mathematics Studies, vol. 147, Princeton University Press, Princeton, NJ, 2000, Hermann Weyl Lectures. The Institute forAdvanced Study.
[Sil96] Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 99, Springer, 1996.
[Tat95] John Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog,Séminaire Bourbaki, Vol. 9, Soc. Math. France, Paris, 1995, pp. Exp. No. 306, 415–
440.
[Wut04] Christian Wuthrich, The fine Selmer group and height pairings, Ph.D. thesis, University of Cambridge, UK, 2004.
, Iwasawa theory of the fine Selmer group, in preparation, 2005.
[Wut05]